A360971 -id:A360971 - cp's OEIS frontend

A360538 Number of multisets of n nonzero digits such that sum(digits) > product(digits).

0, 0, 9, 10, 11, 12, 15, 16, 18, 20, 22, 23, 25, 26, 29, 30, 31, 32, 35, 36, 38, 39, 41, 42, 44, 44, 47, 50, 51, 52, 54, 55, 56, 57, 60, 60, 60, 61, 63, 64, 65, 67, 70, 71, 73, 73, 74, 75, 77, 77, 78, 80, 81, 82, 84, 85, 86, 87, 89, 90, 91, 92, 95, 96, 97, 98, 101, 101

Offset: 0

Charles Bershatsky, Feb 10 2023

Note that this does not represent the number of n-digit numbers that satisfy this property; that would require the computation of the permutations of each multiset.

			For n = 2, the a(2) = 9 solutions are [11,12,13,14,15,16,17,18,19].
For n = 3, the a(3) = 10 solutions are [111,112,113,114,115,116,117,118,119,122].

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A052382, A360971, A360972.

b:= proc(n, i, s, p) option remember; `if`(s+n*i<=p, 0,
     `if`(n=0, 1, add(b(n-1, j, s+j, p*j), j=1..i)))
    end:
a:= n-> b(n, 9, 0, 1):
seq(a(n), n=0..100);  # Alois P. Heinz, Feb 11 2023

A360563 Number of ordered multisets of size n with elements from [n] whose element sum is larger than the product of all elements.

Original entry on oeis.org

0, 0, 3, 10, 31, 71, 171, 288, 505, 985, 1471, 2036, 3455, 5136, 8009, 11376, 14261, 17613, 24073, 34429, 60706, 76196, 92324, 108538, 144947, 167151, 201501, 309115, 452026, 543635, 649137, 928947, 1059705, 1250129, 1634194, 1838908, 2084398, 2331001, 2628518

Offset: 0

Alois P. Heinz, Feb 27 2023

nonn

			a(2) = 3: [1,1], [1,2], [2,1].
a(3) = 10: [1,1,1], [1,1,2], [1,1,3], [1,2,1], [1,2,2], [1,3,1], [2,1,1], [2,1,2], [2,2,1], [3,1,1].
a(4) = 31: [1,1,1,1], [1,1,1,2], [1,1,1,3], [1,1,1,4], [1,1,2,1], [1,1,2,2], [1,1,2,3], [1,1,3,1], [1,1,3,2], [1,1,4,1], [1,2,1,1], [1,2,1,2], [1,2,1,3], [1,2,2,1], [1,2,3,1], [1,3,1,1], [1,3,1,2], [1,3,2,1], [1,4,1,1], [2,1,1,1], [2,1,1,2], [2,1,1,3], [2,1,2,1], [2,1,3,1], [2,2,1,1], [2,3,1,1], [3,1,1,1], [3,1,1,2], [3,1,2,1], [3,2,1,1], [4,1,1,1].

Alois P. Heinz, Table of n, a(n) for n = 0..150

Cf. A360971.

b:= proc(n, i, s, p) option remember;
      `if`(s+n*i<=p, 0, `if`(n=0 or i=1, 1/n!,
       add(b(n-j, i-1, s+i*j, p*i^j)/j!, j=0..n)))
    end:
a:= n-> b(n$2, 0, 1)*n!:
seq(a(n), n=0..44);

cp's OEIS Frontend

A360538 Number of multisets of n nonzero digits such that sum(digits) > product(digits).

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